ARIMA模型在SPSS中的推算过程

1 ARIMAThe ARIMA procedure computes the parameter estimates for a given seasonal or non-seasonal univariate ARIMA model. It also computes the fitted values, forecasting values, and other related variables for the model.NotationThe following notation is used throughout this chapter unless otherwise stated: y t (t =1, 2, ..., N ) Univariate time series under investigationN Total number of observationsa t (t = 1, 2, ... , N ) White noise series normally distributed with mean zero andvariance σa 2pOrder of the non-seasonal autoregressive part of the model qOrder of the non-seasonal moving average part of the model dOrder of the non-seasonal differencing POrder of the seasonal autoregressive part of the model QOrder of the seasonal moving-average part of the model DOrder of the seasonal differencing s Seasonality or period of the modelφp B ()AR polynomial of B of order p , φϕϕϕp p p B B B B ()...=−−−−1122 θq B ()MA polynomial of B of order q , θϑϑϑq q q B B B B ()...=−−−−1122 ΦP B () SAR polynomial of B of order P ,ΦΦΦΦP P P B B B B ()...=−−−−1122ΘQ B ()SMA polynomial of B of order Q , ΘΘΘΘQ Q Q B B B B ()...=−−−−1122 ∇ Non-seasonal differencing operator ∇=−1B∇s Seasonal differencing operator with seasonality s , ∇=−s s B 1BBackward shift operator with By y t t =−1and Ba a t t =−12 ARIMAModelsA seasonal univariate ARIMA(p ,d ,q )(P ,D ,Q )s model is given byφµθp P d s D t q Q t B B y B B a t N ()()[]()(),,ΦΘ∇∇−==1K (1)whereµ is an optional model constant. It is also called the stationary series mean, assuming that, after differencing, the series is stationary. When NOCONSTANT is specified, µ is assumed to be zero.When P Q D ===0, the model is reduced to a (non-seasonal) ARIMA(p ,d ,q ) model:φµθp d t q t B y B a t N ()[](),,∇−==1K (2) An optional log scale transformation can be applied to y t before the model is fitted. In this chapter, the same symbol, y t , is used to denote the series either before or after log scale transformation.Independent variables x 1, x 2, …, x m can also be included in the model. Themodel with independent variables is given byφµθp P d s D t i i m it q Q t B B y c xB B a ()()[()]()()ΦΘ∇∇−−==∑1or ΦΘ()[()()]()B B y c xB a t i i m it t ∇−−==∑1µ (3)whereΦΦ()()()B B B p P =φ∇=∇∇()B d s DΘΘ()()()B B B q Q =θand c i m i ,,,,=12K , are the regression coefficients for the independent variables.ARIMA 3EstimationBasically, two different estimation algorithms are used to compute maximum likelihood (ML) estimates for the parameters in an ARIMA model:• Melard’s algorithm is used for the estimation when there is no missing data inthe time series. The algorithm computes the maximum likelihood estimates of the model parameters. The details of the algorithm are described in Melard (1984), Pearlman (1980), and Morf, Sidhu, and Kailath (1974).•A Kalman filtering algorithm is used for the estimation when some observations in the time series are missing. The algorithm efficiently computes the marginal likelihood of an ARIMA model with missing observations. The details of the algorithm are described in the following literature: Kohn and Ansley (1986) and Kohn and Ansley (1985). Diagnostic StatisticsThe following definitions are used in the statistics below:N p Number of parametersN p q P Q m p q P Q m p =+++++++++%&'without model constant with model constant 1SSQ Residual sum of squaresSSQ =′e e , where e is the residual vector$σa 2 Estimated residual variance$σa SSQ df2=, where df N N p =−SSQ ’ Adjusted residual sum of squaresSSQ SSQ N ’=05Ω1/, where Ω is the theoretical covariancematrix of the observation vector computed at MLE4 ARIMA Log-LikelihoodL N SSQ N a a=−−−ln($)’$ln()σσπ2222 Akaike Information Criterion (AIC)AIC L N p =−+22Schwartz Bayesian Criterion (SBC)SBC L N N p =−+2ln 16Generated VariablesPredicted ValuesForecasting Method: Conditional Least Squares (CLS or AUTOINT)In general, the model used for fitting and forecasting (after estimation, if involved) can be described by Equation (3), which can be written asy D B y B B a c B B x t t t i i m it −=++∇=∑()()()()()ΦΘΦµ1 where D B B B 050505=∇−Φ1ΦΦB 0505µµ=1ARIMA 5Thus, the predicted values (FIT )t are computed as follows:FIT y D B y B B a c B B x t t t t ii m it 05==+++∇=∑$()$()()$()()ΦΘΦµ1 (4)where$$a y y t n t t t =−≤≤1 Starting Values for Computing Fitted Series . To start the computation for fitted values using Equation (4), all unavailable beginning residuals are set to zero and unavailable beginning values of the fitted series are set according to the selected method:• CLS. The computation starts at the (d +sD )-th period. After a specified log scaletransformation, if any, the original series is differenced and/or seasonally differenced according to the model specification. Fitted values for the differenced series are computed first. All unavailable beginning fitted values in the computation are replaced by the stationary series mean, which is equal to the model constant in the model specification. The fitted values are then aggregated to the original series and properly transformed back to the original scale. The first d +sD fitted values are set to missing (SYSMIS ).• AUTOINIT. The computation starts at the [d +p +s (D +P )]-th period. After anyspecified log scale transformation, the actual d +p +s (D +P ) beginning observations in the series are used as beginning fitted values in the computation. The first d +p +s (D +P ) fitted values are set to missing. The fitted values are then transformed back to the original scale, if a log transformation is specified.Forecasting Method: Unconditional Least Squares (EXACT)As with the CLS method, the computations start at the (d+sD)-th period. First, the original series (or the log-transformed series if a transformation is specified) is differenced and/or seasonally differenced according to the model specification. Then the fitted values for the differenced series are computed. The fitted values are one-step-ahead, least-squares predictors calculated using the theoretical autocorrelation function of the stationary autoregressive moving average (ARMA) process corresponding to the differenced series. The autocorrelation function is computed by treating the estimated parameters as the true parameters. The fitted values are then aggregated to the original series and properly transformed back to the original scale. The first d+sD fitted values are set to missing (SYSMIS ). The details of the least-squares prediction algorithm for the ARMA models can be found in Brockwell and Davis (1991).6 ARIMAResidualsResidual series are always computed in the transformed log scale, if a transformation is specified.()(),,,ERR y FIT t N t t t =−=12KStandard Errors of the Predicted ValuesStandard errors of the predicted values are first computed in the transformed log scale, if a transformation is specified.Forcasting Method: Conditional Least Squares (CLS or AUTOINIT)()$,,,SEP t N t a ==σ12KForecasting Method: Unconditional Least Squares (EXACT)In the EXACT method, unlike the CLS method, there is no simple expression for the standard errors of the predicted values. The standard errors of the predicted values will, however, be given by the least-squares prediction algorithm as a byproduct.Standard errors of the predicted values are then transformed back to the original scale for each predicted value, if a transformation is specified.Confidence Limits of the Predicted ValuesConfidence limits of the predicted values are first computed in the transformed log scale, if a transformation is specified:()()(),,,()()(),,,/,/,LCL FIT t SEP t N UCL FIT t SEP t N t t df tt t df t =−==+=−−12121212ααK Kwhere t df 12−α/, is the 12−α/16-th percentile of a t distribution with df degrees of freedom and α is the specified confidence level (by default α=005.).Confidence limits of the predicted values are then transformed back to the original scale for each predicted value, if a transformation is specified.ARIMA 7ForecastingForecasting ValuesForcasting Method: Conditional Least Squares (CLS or AUTOINIT)The following forecasting equation can be derived from Equation (3):$()()$()()$()(),y l D B y B B a c B B x t t l t l ii mi t l =+++∇++=+∑ΦΘΦµ1 (5)whereD B B B B 0505050505=∇−=ΦΦΦ11,µµ $()yl t denotes the l-step-ahead forecast of y t l + at the time t . $$()y y l iy l i l i t l i t l it +−+−=≤−>%&'if if$$()a y y l i l i t l j t l i t l i +−+−+−−=−≤>%&'110if ifNote that $()yt ′1 is the one-step-ahead forecast of y t ′+1 at time ′t , which is exactly the predicted value $yt ′+1as given in Equation (4). Forecasting Method: Unconditional Least Squares (EXACT)The forecasts with this option are finite memory, least-squares forecasts computed using the theoretical autocorrelation function of the series. The details of the least-squares forecasting algorithm for the ARIMA models can be found in Brockwell and Davis (1991).8 ARIMAStandard Errors of the Forecasting ValuesForcasting Method: Conditional Least Squares (CLS or AUTOINIT)For the purpose of computing standard errors of the forecasting values, Equation(1) can be written in the format of ψweights (ignoring the model constant):y a B a B a t t t i it i i q B Q B p B P B ===−=∞∑ϑφψψ()()()()()ΘΦ0, ψ01= (6)Let$()y l t denote the l -step-ahead forecast of y t l +at time t . Then se[$()]{...}$y l t l a =++++−112221212ψψψσ Note that, for the predicted value, l =1. Hence, ()$SEP t a =σat any time t . Computation of ψ Weights . ψ weights can be computed by expanding both sides of the following equation and solving the linear equation system established by equating the corresponding coefficients on both sides of the expansion:φψθp P d s D q Q B B B B B ()()()()()ΦΘ∇∇=An explicit expression of ψ weights can be found in Box and Jenkins (1976).Forecasting Method: Unconditional Least Squares (EXACT)As with the standard errors of the predicted values, the standard errors of the forecasting values are a byproduct during the least-squares forecasting computation. The details can be found in Brockwell and Davis (1991).ReferencesBox, G. E. P., and Jenkins, G. M. 1976. Time series analysis : Forecasting and control . San Francisco: Holden-Day.Brockwell, P. J., and Davis, R. A. 1991. Time series: Theory and methods , 2nd ed. New York: Springer-Verlag.。